Topological metrics as evolutionary and dynamical descriptors of conformational landscapes within protein families

Identifying the key order parameters that connect a protein's native structure to its dynamical and evolutionary behavior remains a central challenge. We introduce topological and geometrical metrics-specifically, writhe and Local Topological Energy (LTE)-to investigate these connections. Applying these tools to both present-day and ancestral forms of thioredoxin and β-lactamase, we show that LTE strongly correlates with established dynamical measures such as the Dynamical Flexibility Index (DFI). Remarkably, LTE distributions also track the evolutionary trajectories of these proteins, suggesting that the topological geometry of the native state encodes key aspects of both dynamics and evolution. Through molecular dynamics simulations, we further reveal critical shifts in the topological landscape of proteins, providing a molecular mechanism by which functional evolution proceeds via alterations in conformational dynamics. Extending our analysis to over 100 proteins, we provide the first compelling evidence that topological descriptors derived from static structures can reliably predict dynamical behavior. In general, our findings demonstrate that simple geometrical metrics capture essential features of protein conformational landscapes, offering a powerful new approach to bridging protein structure, dynamics, and evolution.